Voting and Experimentation with Correlated Types
Bruno Strulovici
Northwestern University
March 24, 2010

Abstract
This note extends Strulovici (2010) to the case where voter types are correlated and/or
ex ante heterogeneous. There are two voters and the risky action is taken according tothe unanimity rule. Increasing correlation between the types of the two voters reduces thenegative payoff externality that voters have on each other, thereby increases experimentationand efficiency. Moreover, it allows voters to learn from each other and, therefore, increases
the speed of experimentation.

1Introduction

In a dynamic voting model where voters gradually learn about their preferences between a“risky” and a “safe” action, Strulovici (2010) identifies two effects, “winner frustration” and“loser trap,” which reduce voters’ incentive to experiment with the risky action, thereby causing
experimentation to be inefficiently brief. In most of that paper,1 preferences are independently
distributed across voters.
This note shows, for the case of two voters and where the risky action requires unanimity, that
increased correlation across voter types alleviates winner frustration and improves efficiency.
Thus, it reduces the negative “payoff externality” that voters have on each other. Moreover,increased correlation creates a positive informational externality, as voters can learn about their
type by observing each other’s payoffs. Ex ante heterogeneity of types also results in moreexperimentation: the more pessimistic voter prefers, other things equal, the other voter to havea higher probability of being a winner, as this reduces the risk of winner frustration. In thelimit, if a voter is a sure winner (i.e., the risky action is surely optimal for him), the other voter
behaves as a single decision maker.2 The intuition for this result is more general: with more ex
ante type heterogeneity, the voter who is pivotal today is more likely to be pivotal in the future
as well (because it is less likely that voter preferences will “cross” one another) and, therefore, isless subject to winner frustration and loser trap effects. Therefore, provided that heterogeneity
is persistent, more heterogeneity today should result in more experimentation.

2Correlation and Heterogeneity

The setting is the same as the benchmark setting of Strulovici (2010), with the following modifications. There are two voters, 1 and 2, who share a common belief about the initial joint
distribution of their types, although this distribution may be asymmetric and exhibit type correlation across voters. Let θ i denote Voter i’s type, and letpϑ1 ϑ2 = P r[(θ 1 , θ 2 ) = (ϑ1 , ϑ2 ),1Section 5 allows for correlation and arbitrary news arrival processes, but does not specifically study the impactof increased correlation on equilibrium and efficiency.
2This case implies zero correlation since one voter has a known type.

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where ϑi ∈ {g, b} describes the possible types (good or bad) of each voter. Also let
pi = P rob[θ i = g]
for i ∈ {1, 2}, and
α=pgg.
p1 p2

α is a measure of the correlation between voter types. The standard correlation measure andα have a one-to-one relationship for any given p1 and p2 . If α = 1, types are uncorrelated. Ingeneral, α takes values in R+ , although not all values of R+ are achievable for given p1 , p2 . For
example, p1 = 1 implies that α = 1, since in that case Voter 1’s type is deterministic hence
uncorrelated with Voter 2’s type. Let ∆ denote the set of pairs (p2 , α) that are achievable aselementary probabilities vary over the four-dimensional simplex, i.e.,

 X

gg
p
ϑ1 ϑ2
ϑ1 ϑ2
∆=
pgg + pbg , gg:
p=1,p≥0.

(p + pgb )(pgg + pbg )ϑ1 ,ϑ2The following proposition is a simple exercise of Bayesian updating, whose proof is easy andomitted.
Proposition 1 (State Dynamics) Beliefs are governed by the following dynamics equations.
When no lump-sum is observed,
gg• dpdt = −λpgg (2 − p1 − p2 )bb

• dpdt = λpbb (p1 + p2 )gbbg

• dpdt = −λpgb (1 − p1 − p2 ), dpdt = −λpbg (1 − p1 − p2 )12• dα
dt = −λα(1 − α)(p + p )When Voter 1 receives a lump-sum,
gb

bb

gb
bg
ppbb
• pbb
+ = 0, p+ = p1 , p+ = 0, p+ = p1

• α+ = 1, p1+ = 1, p2+ = αp2
where the subscript ‘+’ denotes values immediately after the lump-sum is observed, and its
absence denotes values immediately before the lump-sum. Symmetric formulas obtain if instead
Voter 2 receives a lump sum.
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Suppose that R requires unanimity. We assume that the voter who is the less likely of being awinner is in control: if that voter wants to play the risky action, so should the player with ahigher expected type. This assumption is consistent with the elimination of weakly dominatedstrategies. A unanimity equilibrium (UE) is defined as follows: at any time t, if pi ≤ pj , then jvotes for R whenever i does.
Theorem 1 There exists a unique UE. This equilibrium determined by a cut-off function δ :∆ → [0, 1] such that C(p1 , p2 , α) = R if and only if p1 > δ(p2 , α) whenever p1 ≤ p2 , with thereverse relation if p1 > p2 .
Proof.

First suppose that p2 = 1. Then, Voter 1 has full control over the collective decision.

He therefore imposes his optimal policy, which is that of a single decision maker. This definesδ(1, 1) = pSD . This also fully determines the value functions of both voters in that case. Letp 7→ w(p) denote the value function of voter 2, where p is Voter 1’s probability of being awinner is p, and v SD is the value function of a single decision maker, which is also voter 1’svalue function in this case. More generally suppose that at time 0, p10 ≤ p20 . It follows fromProposition 1 that p1t ≤ p2t for all t preceding the first arrival of a lump-sum. In particular,
this implies that 1 has full control of the collective decision (under unanimity) over that period.
Therefore, he chooses a policy θ that solvesZ σ−r(τ −t)1−r(σ−t)21qw(pσ+ ) + (1 − q)v(pσ+ ) ,ut = sup Eedπθτ (τ ) + eθtwhere, letting σi denote the (possibly infinite) time at which i receives his first lump sum,σ = min{σ1 , σ2 } and q = P rob[σ1 < σ2 ]. This is a standard control problem, whose solution isknown to be Markov. Voter 1 is indifferent between R and S at probability level p, if p solves
the equationpg + λp[w(αp2 ) − s/r] + λp2 [v SD (αp) − s/r] = s.

(1)

The left-hand side is increasing in p, equal to 0 for p = 0 and greater than g > s if p = 1.
Therefore, it has a unique root δ(p2 , α). This shows that C(p1 , p2 , α) = R if and only if p1 >δ(p2 , α). The case p1 > p2 obtains by symmetry.
Theorem 2 states that a voter’s incentive to experiment increases both with the other voter’s
probability of being a winner and with voters’ type correlation. Intuitively, if types are morepositively correlated, the risk of winner frustration decreases. The risk of winner frustration is
also lower for a given voter if the other voter is more likely to be a winner. In the extreme case inwhich, say, Voter 2 is a sure winner (i.e., p2 = 1), Voter 1 has full control over collective decisions,and can behave in effect as a single-decision maker. In addition, positive correlation increases
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1

0.9

0.8

0.7

Experimentation

p2

0.6

0.50.4alpha = 10.3
alpha = 30.2

Status quo

0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p1

Figure 1: Experimentation Boundary δ as a function of α. r = 1, λ = 1, s = 1, g = 2.

the speed of learning, which reduces the time cost of experimentation. In the extreme case of
perfect type correlation, the setting is equivalent to one with a single decision maker with twice
the initial learning intensity. Figure 1 shows numerical computations of the experimentation
boundary for several values of the correlation measure α.
Theorem 2 δ is decreasing in both components.
Proof.

The left-hand side of (1), is increasing in p, p2 and α. Therefore, keeping α fixed, theroot δ(p2 , α) must be decreasing in p2 , and similarly keeping p2 fixed, δ(p2 , α) must be decreasingin α.

5References
Strulovici, B. (2010) “Learning While Voting: Determinants of Collective Experimentation,”
forthcoming in Econometrica.

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